problem_id stringdate 1-01-01 00:00:00 100-01-01 00:00:00 | domain listlengths 1 1 | level stringclasses 3
values | tag listlengths 1 1 | reference stringlengths 0 147 | declaration listlengths 0 0 | problem_path stringlengths 16 16 | lean_path stringlengths 25 25 | natural_language_statement stringlengths 62 1.03k | formal_statement stringlengths 109 5.38k |
|---|---|---|---|---|---|---|---|---|---|
0001 | [
"Category"
] | Easy | [
"Basic"
] | [] | problems/0001.md | CAT_statement/S_0001.lean | Theorem: Let $\mathcal{C}$ be a category and $\mathrm{Id}_{\mathcal{C}}$ the identity functor. Then monoid of natural transformations $\mathrm{End}(\mathrm{Id}_{\mathcal{C}})$ is commutative. | import Mathlib
open CategoryTheory
variable {C : Type*} [Category.{v} C]
theorem id_comm (α β : (𝟭 C) ⟶ (𝟭 C)) : α ≫ β = β ≫ α := by
sorry | |
0002 | [
"Category"
] | Easy | [
"Basic"
] | [] | problems/0002.md | CAT_statement/S_0002.lean | Theorem: Let $\mathcal{C}$ be a category and let $f,g$ be morphisms in $\mathcal{C}$ such that $f\circ g$ is monic. Then $g$ is monic. | import Mathlib
open CategoryTheory
variable {C : Type*} [Category C]
theorem monic_of_comp_monic {X Y Z : C} (g : X ⟶ Y) (f : Y ⟶ Z)
[Mono (g ≫ f)] : Mono g := by
sorry | |
0003 | [
"Category"
] | Medium | [
"Basic"
] | [] | problems/0003.md | CAT_statement/S_0003.lean | Theorem: The forgetful functor $\mathcal{T}\mathrm{op}\to\mathcal{S}\mathrm{et}$, $\mathcal{G}\mathrm{rp}\to\mathcal{S}\mathrm{et}$, $\mathcal{R}\mathrm{ing}\to\mathcal{A}\mathrm{b}$, $\mathcal{T}\mathrm{op}_*\to\mathcal{T}\mathrm{op}$ are faithful but not full.\nomenclature{$\mathcal{S}\mathrm{et}$}{the category of se... | import Mathlib
open CategoryTheory Limits
theorem forget_Top_faithful_not_full :
(forget TopCat).Faithful ∧ ¬ (forget TopCat).Full := by
sorry
theorem forget_Grp_faithful_not_full :
(forget Grp).Faithful ∧ ¬ (forget Grp).Full := by
sorry
theorem forget_Ring_Ab_faithful_not_full :
(forget₂ RingCat... | |
0004 | [
"Category"
] | Medium | [
"Basic"
] | [] | problems/0004.md | CAT_statement/S_0004.lean | Theorem: Let $\{*\}\in\mathcal{S}\mathrm{et}$ be the terminal object in $\mathcal{S}\mathrm{et}$.Then $\hom_{\mathcal{S}\mathrm{et}}(\{*\},-):\mathcal{S}\mathrm{et}\to\mathcal{S}\mathrm{et}$ is an equivalence of categories. | import Mathlib
open CategoryTheory
universe u
def fromTerminalFunctor : Type u ⥤ Type u where
obj α := PUnit.{u} → α
map {α β} (f : α → β) := fun g => f ∘ g
map_id := by
intro α
funext g x
rfl
map_comp := by
intro α β γ f g
funext h x
rfl
theorem fromTerminalEquivalence : fromTermin... | |
0005 | [
"Category"
] | Easy | [
"Basic"
] | [] | problems/0005.md | CAT_statement/S_0005.lean | Theorem: Let $\mathcal{C}$ be a category, if every idempotent in $\mathcal{C}$ can be factored into an epimorhisms followed by a monomorphism, then all idempotents split in $\mathcal{C}$. | import Mathlib
open CategoryTheory Idempotents
variable {C : Type*} [Category.{v} C]
theorem idempotent_splitting_from_epi_mono_factorization
(h : ∀ (X : C) (p : X ⟶ X) (hpp : p ≫ p = p),
∃ (Y : C) (e : X ⟶ Y) (he : Epi e) (m : Y ⟶ X) (hm : Mono m), p = e ≫ m) :
IsIdempotentComplete C := by
sorry | |
0006 | [
"Category"
] | Medium | [
"Basic"
] | [] | problems/0006.md | CAT_statement/S_0006.lean | Theorem: Let $\mathcal{C}$ and $\mathcal{D}$ be two categories.
Let $F:\mathcal{C}\to\mathcal{D}$ be a functor.
Then $F$ has a quasi-inverse if and only if
\begin{enumerate}
\item $F$ is fully faithful;
\item $F$ is essentially surjective.
\end{enumerate} | import Mathlib
open CategoryTheory
theorem funtor_has_quasi_inverse_iff {C D : Type*} [Category C] [Category D] (F : C ⥤ D):
(∃ G : D ⥤ C, (Nonempty (Functor.id C ≅ F.comp G)) ∧ (Nonempty (G.comp F ≅ Functor.id D)))
↔ F.IsEquivalence := by
sorry | |
0007 | [
"Category"
] | Medium | [
"Basic"
] | [] | problems/0007.md | CAT_statement/S_0007.lean | Theorem: Let $\mathcal{C}$ be a category and $\mathrm{Kar}(\mathcal{C})$ be its idempotent completion. Let $I:\mathcal{C}\to \mathrm{Kar}(\mathcal{C})$ be the inclusion functor. Then for any category $\mathcal{D}$ in which idempotents split and any functor $F:\mathcal{C}\to \mathcal{D}$, there is a unique (up to isomor... | import Mathlib
open CategoryTheory
variable {C D : Type*} [Category C] [Category D]
theorem karoubi_universal_property [IsIdempotentComplete D] (F : C ⥤ D) :
∃ (F' : Idempotents.Karoubi C ⥤ D), Nonempty ((Idempotents.toKaroubi C) ⋙ F' ≅ F) := by
sorry
theorem karoubi_universal_property_unique_up_to_iso [IsIde... | |
0008 | [
"Category"
] | Medium | [
"Basic"
] | [] | problems/0008.md | CAT_statement/S_0008.lean | Theorem: Let $G_1$ and $G_2$ be two objects in the category $\mathcal{G}\mathrm{rp}$ of groups.\nomenclature{$\mathcal{G}\mathrm{rp}$}{the category of groups}
The coproduct of $G_1$ and $G_2$ in $\mathcal{G}\mathrm{rp}$ is equivalent to the free product of $G_1$ and $G_2$. | import Mathlib
open CategoryTheory Limits
universe u
variable {G H : Grp.{u}}
theorem freeProdGrp_iso_coprod [HasBinaryCoproduct G H] :
Nonempty (Monoid.Coprod G H ≅ coprod G H) := by
sorry | |
0009 | [
"Category"
] | Medium | [
"Basic"
] | [] | problems/0009.md | CAT_statement/S_0009.lean | Theorem: There exists a morphism in $\mathcal{R}\mathrm{ing}$ such that it is epic but not surjective.\nomenclature{$\mathcal{R}\mathrm{ing}$}{the category of rings} | import Mathlib
open CategoryTheory
theorem exists_epic_not_surjective_in_Ring :
∃ (A B : RingCat) (f : A ⟶ B), Epi f ∧ ¬ Function.Surjective f := by
sorry | |
0010 | [
"Category"
] | Medium | [
"Basic"
] | [] | problems/0010.md | CAT_statement/S_0010.lean | Theorem: Let $F:\mathcal{G}\mathrm{rp}\to \mathcal{S}\mathrm{et}$ be the functor that $G\mapsto \{g\in G\mid g^2=1\}$.
Then $F$ is representable. | import Mathlib
open CategoryTheory
def functor_involution : Grp.{u} ⥤ Type u where
obj := fun G => { g : G.carrier | g * g = 1 }
map := fun {G H} f x => ⟨f.hom x.val, by
refine Set.mem_setOf.mpr ?_
rcases x with ⟨g, hg⟩
simp only [Set.mem_setOf_eq] at hg
rw [← f.hom.map_mul, hg]
simp only [ma... | |
0011 | [
"Category"
] | High | [
"Basic"
] | Exercise 9D, Adámek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0011.md | CAT_statement/S_0011.lean | Definition: Let $(\mathcal C,U)$ be a concrete category over $\mathcal B$.
A morphism $f: x\to y$ in $\mathcal C$ is called $\textbf{initial}$ if for any object $c\in \mathcal C$, a morphism $g:U(c)\to U(x)$ is a morphism in $\mathcal{C}$ whenever $f\circ g: U(c)\to U(y)$ is a morphism in $\mathcal C$.
Definition: An... | import Mathlib
open CategoryTheory
namespace CAT_statement_S_0011
universe u uX
variable {X : Type uX} [Category.{vX} X]
namespace AHS
structure ConcreteCat (X : Type v) [Category X] where
C : Type u
[cat : Category C]
U : C ⥤ X
[U_Faithful : U.Faithful]
attribute [instance] ConcreteCat.cat ConcreteCat.U... |
0012 | [
"Category"
] | High | [
"Basic"
] | Exercise 7P, Adámek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0012.md | CAT_statement/S_0012.lean | Theorem: Let $\mathcal{C}$ be a category and let $f:x\to y$ be a morphism in $\mathcal{C}$.
Then $f$ is a monomorphism in $\mathcal{C}$ if and only if there exists a category $\mathcal{D}$ and a faithful functor $I:\mathcal{C}\to\mathcal{D}$ such that $f$ is a section in $\mathcal{D}$. | import Mathlib
open CategoryTheory Functor
theorem mono_iff_exists_embedding_section
{C : Type u} [Category.{v} C] {X Y : C} (f : X ⟶ Y) :
Mono f ↔ ∃ (D : Type (max u v)) (_ : Category.{v} D) (I : C ⥤ D) (_ : Faithful I),
IsSplitMono (I.map f) := by
sorry |
0013 | [
"Category"
] | High | [
"Basic"
] | 3.39 EXAMPLES (6), Adámek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0013.md | CAT_statement/S_0013.lean | Theorem: The category $\mathcal{T}\mathrm{op}^{CH}$ of compact Hausdorff space is dually equivalent to the category of commutative unital $C^*$-algebras and algebra homomorphisms.\nomenclature{$\mathcal{T}\mathrm{op}^{CH}$}{the category of compact Hausdorff topological spaces} | import Mathlib
open CategoryTheory
universe u
structure CommCStarAlgCat : Type (u + 1) where
of ::
carrier : Type u
[commCStarAlgebra : CommCStarAlgebra carrier]
attribute [instance] CommCStarAlgCat.commCStarAlgebra
namespace CommCStarAlgCat
instance : CoeSort CommCStarAlgCat (Type u) :=
⟨CommCStarAlgCa... |
0014 | [
"Category"
] | High | [
"Basic"
] | 9.3 EXAMPLES (2), Adámek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0014.md | CAT_statement/S_0014.lean | Definition: Let $(\mathcal C,U)$ be a concrete category over $\mathcal B$.
A morphism $f: x\to y$ in $\mathcal C$ is called $\textbf{initial}$ if for any object $c\in \mathcal C$, a morphism $g:U(c)\to U(x)$ is a morphism in $\mathcal{C}$ whenever $f\circ g: U(c)\to U(y)$ is a morphism in $\mathcal C$.
Definition: An... | import Mathlib
open CategoryTheory
namespace CAT_statement_S_0014
universe u uX
variable {X : Type uX} [Category.{vX} X]
namespace AHS2
structure ConcreteCat (X : Type v) [Category X] where
C : Type u
[cat : Category C]
U : C ⥤ X
[U_Faithful : U.Faithful]
attribute [instance] ConcreteCat.cat ConcreteCat.... |
0015 | [
"Category"
] | High | [
"Basic"
] | 9.3 EXAMPLES (2)(b), Adámek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0015.md | CAT_statement/S_0015.lean | Definition: Let $(\mathcal C,U)$ be a concrete category over $\mathcal B$.
A morphism $f: x\to y$ in $\mathcal C$ is called $\textbf{initial}$ if for any object $c\in \mathcal C$, a morphism $g:U(c)\to U(x)$ is a morphism in $\mathcal{C}$ whenever $f\circ g: U(c)\to U(y)$ is a morphism in $\mathcal C$.
Definition: An... | import Mathlib
open CategoryTheory
namespace CAT_statement_S_0015
universe u uX
variable {X : Type uX} [Category.{vX} X]
namespace AHS
structure ConcreteCat (X : Type v) [Category X] where
C : Type u
[cat : Category C]
U : C ⥤ X
[U_Faithful : U.Faithful]
attribute [instance] ConcreteCat.cat ConcreteCat.U... |
0016 | [
"Category"
] | High | [
"Basic"
] | 9.3 EXAMPLES (3)(b), Adámek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0016.md | CAT_statement/S_0016.lean | Definition: Let $(\mathcal C,U)$ be a concrete category over $\mathcal B$.
A morphism $f: x\to y$ in $\mathcal C$ is called $\textbf{initial}$ if for any object $c\in \mathcal C$, a morphism $g:U(c)\to U(x)$ is a morphism in $\mathcal{C}$ whenever $f\circ g: U(c)\to U(y)$ is a morphism in $\mathcal C$.
Definition: An... | import Mathlib
open CategoryTheory
namespace CAT_statement_S_0016
universe u uX
variable {X : Type uX} [Category.{vX} X]
namespace AHS
structure ConcreteCat (X : Type v) [Category X] where
C : Type u
[cat : Category C]
U : C ⥤ X
[U_Faithful : U.Faithful]
attribute [instance] ConcreteCat.cat ConcreteCat.U... |
0017 | [
"Category"
] | High | [
"Basic"
] | 9.3 EXAMPLES (4)(d), Adámek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0017.md | CAT_statement/S_0017.lean | Definition: Let $(\mathcal C,U)$ be a concrete category over $\mathcal B$.
A morphism $f: x\to y$ in $\mathcal C$ is called $\textbf{initial}$ if for any object $c\in \mathcal C$, a morphism $g:U(c)\to U(x)$ is a morphism in $\mathcal{C}$ whenever $f\circ g: U(c)\to U(y)$ is a morphism in $\mathcal C$.
Definition: An... | import Mathlib
open CategoryTheory Limits TopologicalSpace
namespace CAT_statement_S_0017
universe u uX
variable {X : Type uX} [Category.{vX} X]
namespace AHS
structure ConcreteCat (X : Type v) [Category X] where
C : Type u
[cat : Category C]
U : C ⥤ X
[U_Faithful : U.Faithful]
attribute [instance] Concr... |
0018 | [
"Category"
] | High | [
"Basic"
] | 9.3 EXAMPLES (4)(a), Adámek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0018.md | CAT_statement/S_0018.lean | Definition: Let $(\mathcal C,U)$ be a concrete category over $\mathcal B$.
A morphism $f: x\to y$ in $\mathcal C$ is called $\textbf{initial}$ if for any object $c\in \mathcal C$, a morphism $g:U(c)\to U(x)$ is a morphism in $\mathcal{C}$ whenever $f\circ g: U(c)\to U(y)$ is a morphism in $\mathcal C$.
Definition: An... | import Mathlib
open CategoryTheory
namespace CAT_statement_S_0018
universe u uX
variable {X : Type uX} [Category.{vX} X]
namespace AHS
structure ConcreteCat (X : Type v) [Category X] where
C : Type u
[cat : Category C]
U : C ⥤ X
[U_Faithful : U.Faithful]
attribute [instance] ConcreteCat.cat ConcreteCat.U... |
0019 | [
"Category"
] | Easy | [
"Adjunction"
] | [] | problems/0019.md | CAT_statement/S_0019.lean | Theorem: A functor $G : \mathcal{D} \to \mathcal{C}$ has a left adjoint if and only if for each $c \in\mathcal{C}$, the comma category $(c \downarrow G)$ has an initial object. | import Mathlib
open CategoryTheory Limits
variable {C : Type*} {D : Type*} [Category.{v₁} C] [Category.{v₂} D]
theorem functor_hasLeftAdjoint_iff_structuredArrow_hasInitial
(G : D ⥤ C) :
G.IsRightAdjoint ↔ ∀ c : C, HasInitial (StructuredArrow c G) := by
sorry | |
0020 | [
"Category"
] | Easy | [
"Adjunction"
] | [] | problems/0020.md | CAT_statement/S_0020.lean | Theorem: Let $\mathcal{C}$ and $\mathcal{D}$ be categories and let $F:\mathcal{C}\to \mathcal{D}$ be a functor that admits a right adjoint $G$.
Then $F$ is fully faithful if and only if $u:\mathrm{Id}_{\mathcal{C}}\to G\circ F$ is isomorphism. | import Mathlib
open CategoryTheory
variable {C D : Type*} [Category C] [Category D] (F : C ⥤ D) (G : D ⥤ C)
theorem fully_faithful_iff_unit_isIso (adj : F ⊣ G) :
(F.Full ∧ F.Faithful) ↔ IsIso adj.unit := by
sorry | |
0021 | [
"Category"
] | Medium | [
"Adjunction"
] | [] | problems/0021.md | CAT_statement/S_0021.lean | Theorem: Let $\mathcal{C}$ and $\mathcal{D}$ be categories and let $F:\mathcal{C}\to \mathcal{D}$ be a functor that admits a right adjoint $G$.
Then $G$ is an equivalence of categories if and only if $F$ is fully faithful and $G$ is conservative. | import Mathlib
open CategoryTheory
variable {C : Type u₁} [Category.{v₁} C] {D : Type u₂} [Category.{v₂} D]
theorem right_adjoint_isEquivalence_iff_left_full_faithful_and_right_conservative
(F : C ⥤ D) (G : D ⥤ C) (adj : F ⊣ G) :
G.IsEquivalence ↔ (F.Full ∧ F.Faithful) ∧ G.ReflectsIsomorphisms := by
sorry | |
0022 | [
"Category"
] | Medium | [
"Adjunction"
] | [] | problems/0022.md | CAT_statement/S_0022.lean | Theorem: Let $\mathcal{C}$ and $\mathcal{D}$ be locally small categories and let $F:\mathcal{C}\to \mathcal{D}$ be a functor.
Then $F$ admits a right adjoint if and only if for each $d\in\mathcal{D}$, $\hom_{\mathcal{D}}(F(-),d):\mathcal{C}^{op}\to\mathcal{S}\mathrm{et}$ is representable. | import Mathlib
open CategoryTheory
variable {C : Type u₁} [Category.{v} C] {D : Type u₂} [Category.{v} D]
theorem isLeftAdjoint_iff_yoneda_comp_op_isRepresentable (F : C ⥤ D) :
F.IsLeftAdjoint ↔ ∀ (d : D), (F.op ⋙ yoneda.obj d).IsRepresentable := by
sorry | |
0023 | [
"Category"
] | Medium | [
"Adjunction"
] | [] | problems/0023.md | CAT_statement/S_0023.lean | Theorem: Let $A,B$ be rings and $\phi : A \to B$ be a ring homomorphism. The functor $\phi_* : B-\mathcal{M}\mathrm{od} \to A-\mathcal{M}\mathrm{od}$ between their module categories is defined by $(N,l_N) \mapsto (N,l_N \circ (\phi \otimes \mathrm{id}))$. Then the functor $\phi_*$ admits a left adjoint $\phi^* := B \ot... | import Mathlib
open CategoryTheory
theorem ring_hom_induced_functor_has_adjoints
{A B : RingCat} (φ : A ⟶ B) :
∃ (φ_pull : ModuleCat B ⥤ ModuleCat A)
(φ_push : ModuleCat A ⥤ ModuleCat B)
(φ_coind : ModuleCat A ⥤ ModuleCat B),
Nonempty (Adjunction φ_push φ_pull) ∧ Nonempty (Adjunction φ_pull ... | |
0024 | [
"Category"
] | Easy | [
"Adjunction"
] | [] | problems/0024.md | CAT_statement/S_0024.lean | Theorem: The forgetful functor $U : \mathcal{A}\mathrm{b} \to \mathcal{G}\mathrm{rp}$ admits a left adjoint. | import Mathlib
open CategoryTheory
universe u
theorem forget_CommGrp_to_Grp_admits_left_adjoint :
(forget₂ CommGrp.{u} Grp.{u}).IsRightAdjoint := by
sorry | |
0025 | [
"Category"
] | Medium | [
"Adjunction"
] | [] | problems/0025.md | CAT_statement/S_0025.lean | Theorem: Let $\mathcal{C}$, $\mathcal{D}$, $\mathcal{E}$ be categories and $U : \mathcal{D}\to \mathcal{C}$, $V : \mathcal{E}\to \mathcal{C}$, $F : \mathcal{D} \to \mathcal{E}$ be functors such that $V \circ F = U$. Suppose $U, V$ have left adjoints and $\mathcal D$ admits coequalizers. If $V$ reflects split epimorphis... | import Mathlib
open CategoryTheory
variable {C D E : Type*} [Category C] [Category D] [Category E]
namespace CategoryTheory
class Functor.ReflectsSplitEpimorphismsToRegularEpimorphisms (F : Functor C D) : Prop where
reflects : ∀ {X Y} {f : X ⟶ Y} [IsSplitEpi (F.map f)], Nonempty (RegularEpi f)
end CategoryTheor... | |
0026 | [
"Category"
] | Medium | [
"Adjunction"
] | Exercise 19I, Adámek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0026.md | CAT_statement/S_0026.lean | Theorem: Let $F,G,H$ be functors such that $F\dashv G\dashv H$.
Then $F$ is fully faithful if and only if $H$ is fully faithful. | import Mathlib
open CategoryTheory Functor
variable {C : Type u₁} [Category.{v₁} C] {D : Type u₂} [Category.{v₂} D]
variable {F : C ⥤ D} {G : D ⥤ C} {H : C ⥤ D}
theorem fullyFaithful_iff_of_adjoints (hFG : F ⊣ G) (hGH : G ⊣ H) :
(F.Full ∧ F.Faithful) ↔ (H.Full ∧ H.Faithful) := by
sorry |
0027 | [
"Category"
] | Medium | [
"Adjunction"
] | Exercise 18H, Adámek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0027.md | CAT_statement/S_0027.lean | Theorem: Let $(\mathbb{Z},\leq)$ be a poset, regarded as a category, then $f\in \mathrm{End}(\mathbb{Z})$ has left adjoint if and only if it has a right adjoint. | import Mathlib
open CategoryTheory
theorem int_endofunctor_hasLeftAdjoint_iff_hasRightAdjoint (f : ℤ ⥤ ℤ) :
f.IsRightAdjoint ↔ f.IsLeftAdjoint := by
sorry |
0028 | [
"Category"
] | Medium | [
"Adjunction"
] | Exercise 19F (a), Adámek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0028.md | CAT_statement/S_0028.lean | Theorem: View the poset $\mathbb{N} = (\mathbb{N},\leq)$ of natural numbers as a category. There is a sequence of distinct functors $G_n:\mathbb{N}\to \mathbb{N}$ such that $G_0(x)=x+1$ and $G_{n+1}\dashv G_n$ for each $n\in \mathbb{N}$. | import Mathlib
open CategoryTheory
theorem exists_sequence_of_distinct_adjoints_nat :
∃ G : ℕ → (ℕ ⥤ ℕ),
Function.Injective G ∧
(∀ x, (G 0).obj x = x + 1) ∧
(∀ n, Nonempty (G (n + 1) ⊣ G n)) := by
sorry |
0029 | [
"Category"
] | Medium | [
"Adjunction"
] | [] | problems/0029.md | CAT_statement/S_0029.lean | Theorem: Let $(-)^{\times}: \mathcal{R}\mathrm{ing} \to \mathcal{G}\mathrm{rp}$ mapping a ring to its group of units. Then it admits a left adjoint. | import Mathlib
open CategoryTheory
def RingCat.units : RingCat.{u} ⥤ Grp.{u} where
obj R := .of Rˣ
map f := Grp.ofHom (Units.map f.hom)
theorem exists_leftAdjoint_unitFunctor :
∃ (left : Grp.{u} ⥤ RingCat.{u}), Nonempty (left ⊣ RingCat.units.{u}) := by
sorry | |
0030 | [
"Category"
] | High | [
"Reflective"
] | Exercise 4G(d), Adámek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0030.md | CAT_statement/S_0030.lean | Theorem: There are categories $\mathcal{C}$, $\mathcal{D}$ and $\mathcal{E}$ such that $\mathcal{C}$ is a subcategory of $\mathcal{D}$, $\mathcal{D}$ is a subcategory of $\mathcal{E}$ and $\mathcal{C}$ is reflective in $\mathcal{E}$, but $\mathcal{C}$ is not reflective in $\mathcal{D}$. Here we do not require a reflect... | import Mathlib
open CategoryTheory Functor
universe u v
namespace CategoryTheory
open Category Adjunction
variable {C : Type u₁} {D : Type u₂} {E : Type u₃}
variable [Category.{v₁} C] [Category.{v₂} D] [Category.{v₃} E]
class Reflective2 (R : D ⥤ C) extends R.Faithful where
L : C ⥤ D
adj : L ⊣ R
end Category... |
0031 | [
"Category"
] | High | [
"Reflective"
] | Exercise 4E(d), Adámek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0031.md | CAT_statement/S_0031.lean | Theorem: Neither the category $\mathcal{S}\mathrm{et}$ of sets nor the category $\mathcal{T}\mathrm{op}$ of topological spaces has a proper isomorphism-closed full subcategory that is both reflective and coreflective. | import Mathlib
open CategoryTheory
theorem not_reflective_and_coreflective (P : ObjectProperty (Type u))
(h : P.IsClosedUnderIsomorphisms) (hproper : ∃ X : Type u, ¬ P X) :
IsEmpty (Reflective P.ι) ∨ IsEmpty (Coreflective P.ι) := by
sorry |
0032 | [
"Category"
] | High | [
"Reflective"
] | Exercise 4D(a), Adámek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0032.md | CAT_statement/S_0032.lean | Theorem: $\mathcal{S}\mathrm{et}$ has precisely three full, isomorphism-closed, reflective subcategories. | import Mathlib
open CategoryTheory Functor Limits
namespace CAT_statement_S_0032
def IsIsoClosed (P : Type u → Prop) : Prop :=
∀ {X Y : Type u}, Nonempty (X ≅ Y) → P X → P Y
def SubcategoryEquiv (P Q : Type u → Prop) : Prop :=
∀ X, P X ↔ Q X
def IsReflectiveSubcategory (P : Type u → Prop) : Prop :=
Nonempty ... |
0033 | [
"Category"
] | High | [
"Reflective"
] | Exercise 4D(b), Adámek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0033.md | CAT_statement/S_0033.lean | Theorem: $\mathcal{T}\mathrm{op}^{CH}$ has precisely two full, isomorphism-closed, coreflective subcategories. | import Mathlib
open CategoryTheory Topology
namespace CAT_statement_S_0033
structure FullCoreflectiveSubcategory (C : Type u) [Category.{v} C] where
obj : ObjectProperty C
iso_closed : obj.IsClosedUnderIsomorphisms
coreflective : Coreflective obj.ι
theorem CompHaus_has_precisely_two_coreflective_subcategories... |
0034 | [
"Category"
] | Medium | [
"Concrete"
] | Exercise 10R, Adámek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0034.md | CAT_statement/S_0034.lean | Definition: Let $(\mathcal C, U)$ be a concrete category over $\mathcal B$.
A $\textbf{universal arrow}$ over $x\in \mathcal B$ is a structured arrow $u:x\to U(c)$ with domain $x$ that has the following universal property: for each structured arrow $f: x\to U(b)$ with domain $x$ there exists a unique morphism $\underl... | import Mathlib
open CategoryTheory Limits Functor Opposite
namespace CAT_statement_S_0034
variable {C : Type u} [Category.{v} C]
def IsFreeObject (U : C ⥤ Type v) (d : C) (I : Type v) : Prop :=
∃ (η : I ⟶ U.obj d), ∀ {y : C} (f : I ⟶ U.obj y), ∃! (g : d ⟶ y), U.map g ∘ η = f
def IsCopower (x d : C) (I : Type v) ... |
0035 | [
"Category"
] | High | [
"Concrete"
] | Exercise 5O(k), Adámek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0035.md | CAT_statement/S_0035.lean | Definition: A full concrete embedding is called a realization.
Theorem: There is a construct $(\mathcal{C},U)$ such that every construct has a realization to $(\mathcal{C},U)$. | import Mathlib
open CategoryTheory
namespace CAT_statement_S_0035
structure Construct where
C : Type u
[str : Category.{v} C]
U : C ⥤ Type u
[faithful : Functor.Faithful U]
attribute [instance] Construct.str Construct.faithful
def IsRealization (S T : Construct.{u, v}) (F : S.C ⥤ T.C) : Prop :=
F ⋙ T.U =... |
0036 | [
"Category"
] | High | [
"Concrete"
] | Exercise 5J(d), Adámek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0036.md | CAT_statement/S_0036.lean | Definition: A category $\mathcal C$ is called $\textbf{concretizable}$ over a category $\mathcal B$ if there exists a faithful functor $U:\mathcal C\to \mathcal B$.
Theorem: There exist categories that are not concretizable over $\mathcal{S}\mathrm{et}$. | import Mathlib
open CategoryTheory
theorem exists_category_not_concretizable_over_Type :
∃ (C : Type u) (_ : Category.{v} C), ¬ ∃ (F : C ⥤ Type v), F.Faithful := by
sorry |
0037 | [
"Category"
] | High | [
"Concrete"
] | Exercise 5D(d), Adámek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0037.md | CAT_statement/S_0037.lean | Theorem: There are precisely two concrete functors from the category $\mathcal{S}\mathrm{et}$ of sets to the category $\mathcal{T}\mathrm{op}$ of topological spaces, but a proper class of concrete functors from $\mathcal{T}\mathrm{op}$ into itself. | import Mathlib
open CategoryTheory
namespace CAT_statement_S_0037
universe u v w
variable {X : Type uX} [Category.{vX} X]
structure ConcreteCat (X : Type v) [Category X] where
C : Type u
[cat : Category C]
U : C ⥤ X
[U_Faithful : U.Faithful]
attribute [instance] ConcreteCat.cat ConcreteCat.U_Faithful
de... |
0038 | [
"Category"
] | High | [
"Concrete"
] | Exercise 10L, Adámek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0038.md | CAT_statement/S_0038.lean | Definition: Let $\mathcal C$ be a category and let $c\in \mathcal C$ be an object.
A $\textbf{regular subobject}$ of $c$ is a pair $(x,i)$ where $i$ is a regular monomorphism.
Definition: Let $\mathcal C$ be a category.
$\mathcal C$ is called $\textbf{regular wellpowered}$ if no object in $\mathcal C$ has a proper cla... | import Mathlib
namespace CAT_statement_S_0038
open CategoryTheory Limits
universe u v w
variable {C : Type u} [Category C] [HasFiniteLimits C]
def IsConcretizable (X : Type v) [Category X] (D: Type u) [Category D] : Prop :=
∃ (U : D ⥤ X), U.Faithful
variable (C)
class RegularWellPowered : Prop where
regularS... |
0039 | [
"Category"
] | High | [
"Concrete"
] | Exercise 5L, Adámek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0039.md | CAT_statement/S_0039.lean | Definition: A $\textbf{construct}$ is a concrete category over the category $\mathcal{S}\mathrm{et}$ of sets.
Theorem: Let $\mathcal{F}\mathrm{rm}$ be the construct of frames (i.e., distributive suplattices) and frame homomorphisms. Then there is a unique concrete functor $T: \mathcal{T}\mathrm{op}^{op}_0 \to \mathcal... | import Mathlib
open CategoryTheory Topology
universe u v w
variable {X : Type uX} [Category.{vX} X]
namespace CAT_statement_S_0039
structure ConcreteCat (X : Type v) [Category X] where
C : Type u
[cat : Category C]
U : C ⥤ X
[U_Faithful : U.Faithful]
attribute [instance] ConcreteCat.cat ConcreteCat.U_Fait... |
0040 | [
"Category"
] | High | [
"Concrete"
] | Exercise 8G(a), Adámek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0040.md | CAT_statement/S_0040.lean | Definition: Let $(\mathcal C, U)$ be a concrete category over $\mathcal B$.
A $\textbf{universal arrow}$ over $x\in \mathcal B$ is a structured arrow $u:x\to U(c)$ with domain $x$ that has the following universal property: for each structured arrow $f: x\to U(b)$ with domain $x$ there exists a unique morphism $\underl... | import Mathlib
open CategoryTheory
namespace CAT_statement_S_0040
universe u v w
variable {X : Type uX} [Category.{vX} X]
structure ConcreteCat (X : Type v) [Category X] where
C : Type u
[cat : Category C]
U : C ⥤ X
[U_Faithful : U.Faithful]
attribute [instance] ConcreteCat.cat ConcreteCat.U_Faithful
ab... |
0041 | [
"Category"
] | High | [
"Concrete"
] | Exercise 8G(b), Adámek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0041.md | CAT_statement/S_0041.lean | Definition: Let $(\mathcal C, U)$ be a concrete category over $\mathcal B$. A $\textbf{universal arrow}$ over $x \in \mathcal B$ is a morphism $u:x\to U(c)$ that has the following universal property: for each morphism $f: x\to U(b)$ there exists a unique morphism $f' : c\to b$ such that $U(f') \circ u = f$.
Definition... | import Mathlib
open CategoryTheory
universe u v w
namespace CAT_statement_S_0041
structure FreeObject {C : Type u} [Category.{v} C] [HasForget.{w} C] (x : Type w) where
(obj : C)
(emb : x ⟶ (forget C).obj obj)
(uniq : ∀ (Y : C) (f : x ⟶ (forget C).obj Y), ∃! (g : obj ⟶ Y), emb ≫ (forget C).map g = f)
theore... |
0042 | [
"Category"
] | Easy | [
"Limit"
] | [] | problems/0042.md | CAT_statement/S_0042.lean | Theorem: Let $\mathcal{C}$ and $\mathcal{D}$ be categories and let $F:\mathcal{C}\to \mathcal{D}$ be a fully faithful functor.
Then $F$ reflects any limits and colimits admitted in the codomain category. | import Mathlib
open CategoryTheory Limits Functor
variable {C : Type u₁} [Category.{v₁} C] {D : Type u₂} [Category.{v₂} D]
theorem fully_faithful_reflects_limits_and_colimits (F : C ⥤ D) [Full F] [Faithful F] :
ReflectsLimits F ∧ ReflectsColimits F := by
sorry | |
0043 | [
"Category"
] | Easy | [
"Limit"
] | [] | problems/0043.md | CAT_statement/S_0043.lean | Theorem: The one point set $\{*\}$ form a separator in Set, and the two point set $\{a,b\}$ form a coseparating set in Set. | import Mathlib
open CategoryTheory Function Classical
theorem PUnit_isSeparator : IsSeparator (PUnit : Type u) := by
sorry
theorem ULiftBool_isCoseparator : IsCoseparator (ULift.{u} Bool) := by
sorry | |
0044 | [
"Category"
] | Easy | [
"Limit"
] | [] | problems/0044.md | CAT_statement/S_0044.lean | Theorem: Filtered colimits commute with finite limits in $\mathcal{S}\mathrm{et}$. | import Mathlib
open CategoryTheory Limits
variable {J : Type u} [SmallCategory J] [FinCategory J]
variable {K : Type u} [SmallCategory K] [IsFiltered K]
variable (F : J ⥤ K ⥤ Type u)
theorem filteredColimitsCommuteWithFiniteLimits :
Nonempty (colimit (limit F) ≅ limit (colimit F.flip)) := by
sorry | |
0045 | [
"Category"
] | Easy | [
"Limit"
] | [] | problems/0045.md | CAT_statement/S_0045.lean | Theorem: Let $\omega$ be the ordinal of natural numbers.
Consider $F:\omega^{op}\to \mathcal{R}\mathrm{ing}$ with $F_n:=\mathbb{Z}/p^n\mathbb{Z}$ and $f_n:F_{n+1}\to F_n$.
Then the limit exists. | import Mathlib
open CategoryTheory Limits Opposite
variable (p : ℕ)
noncomputable def pAdicFunctor : ℕᵒᵖ ⥤ RingCat where
obj n := RingCat.of (ZMod (p ^ (unop n)))
map {m n} f := RingCat.ofHom <|
ZMod.castHom (pow_dvd_pow p (leOfHom f.unop)) (ZMod (p ^ (unop n)))
map_id := by
intro n
ext x
simp... | |
0046 | [
"Category"
] | Easy | [
"Limit"
] | [] | problems/0046.md | CAT_statement/S_0046.lean | Theorem: Let $\mathcal{C}$ and $\mathcal{D}$ be a small category and let $F,G:\mathcal{C}\to\mathcal{D}$ be two functors.
Then we have $\mathrm{Nat}(F,G)\cong \int_{c\in\mathcal{C}}\hom_{\mathcal{D}}(F(c),G(c))$. | import Mathlib
open CategoryTheory Limits
variable {C : Type u} [SmallCategory C]
variable {D : Type u} [SmallCategory D]
variable (F G : C ⥤ D)
def homIntegrandBifunctor : Cᵒᵖ × C ⥤ Type u :=
(Functor.prod F.op G) ⋙ (Functor.hom D)
theorem natTransIsoEnd :
Nonempty (NatTrans F G ≅ end_ (curryObj (homIntegra... | |
0047 | [
"Category"
] | Easy | [
"Limit"
] | [] | problems/0047.md | CAT_statement/S_0047.lean | Theorem: There is no equivalence of categories between $\mathcal{S}\mathrm{et}$ and $\mathcal{S}\mathrm{et}^{op}$. | import Mathlib
open CategoryTheory
theorem no_equiv_between_Set_and_op : ¬ Nonempty (Equivalence (Type u) (Type u)ᵒᵖ) := by
sorry | |
0048 | [
"Category"
] | Easy | [
"Limit"
] | [] | problems/0048.md | CAT_statement/S_0048.lean | Theorem: A reflective subcategory $\mathcal{C}$ of a cocomplete category $\mathcal{D}$ is also cocomplete. | import Mathlib
open CategoryTheory Limits
variable {C : Type u} [Category.{v} C] {D : Type u} [Category.{v} D]
theorem hasColimits_of_reflective (i : C ⥤ D) [Reflective i] [HasColimits D] :
HasColimits C := by
sorry | |
0049 | [
"Category"
] | Easy | [
"Limit"
] | [] | problems/0049.md | CAT_statement/S_0049.lean | Theorem: Let $\mathcal{C}$ and $\mathcal{E}$ be two categories and let $F:\mathcal{C}\to \mathcal{E}$ be a functor.
Let $\bullet $ be the terminal category consisting of a unique object $\bullet$ and a unique morphism.
Then a colimit of $F$ is a left Kan extension of $F$ along $K:\mathcal{C}\to \bullet$, i.e. $... | import Mathlib
open CategoryTheory Limits
universe u₁ v₁ u₂ v₂
variable {C : Type u₁} [Category.{v₁} C]
variable {E : Type u₂} [Category.{v₂} E]
theorem colimit_is_leftKanExtension_along_to_terminal
(F : C ⥤ E) (K : C ⥤ PUnit) [HasColimit F] [K.HasLeftKanExtension F] :
Nonempty ((K.leftKanExtension F).obj ... | |
0050 | [
"Category"
] | Medium | [
"Limit"
] | [] | problems/0050.md | CAT_statement/S_0050.lean | Definition: Let $\mathcal C$ be a locally small category.
An object $c \in\mathcal C$ is called $\textbf{compact}$ if $\mathrm{hom}_{\mathcal C} (c,-)$ preserves filtered colimits.
Theorem: For $\mathcal{S}\mathrm{et}$, an object is compact if and only if it is a finite set. | import Mathlib
open CategoryTheory Limits
theorem isCompactObject_iff_finite_type (X : Type u) :
PreservesFilteredColimits (coyoneda.obj (Opposite.op X)) ↔ Finite X := by
sorry | |
0051 | [
"Category"
] | Medium | [
"Limit"
] | [] | problems/0051.md | CAT_statement/S_0051.lean | Theorem: Let $\mathcal{C}$ be a category.
Then $\mathcal{C}$ admits all small limits if and only if $\mathcal{C}$ admits all small products and pullbacks. | import Mathlib
open CategoryTheory Limits
variable {C : Type u} [Category.{v} C]
theorem has_limits_iff_has_products_and_pullbacks :
HasLimitsOfSize.{v, v} C ↔ (∀ (J : Type v), HasLimitsOfShape (Discrete J) C) ∧ HasLimitsOfShape WalkingCospan C := by
sorry | |
0052 | [
"Category"
] | Medium | [
"Limit"
] | [] | problems/0052.md | CAT_statement/S_0052.lean | Theorem: Let $X,Y,Z$ be objects in $\mathcal{S}\mathrm{et}$ with morphisms $f:X\to Z$ and $g:Y\to Z$.
Then $\{(x,y)\in X\times Y\mid f(x)=g(y)\}$ is the pullback $X\times_Z Y$ of $X$ and $Y$ over $Z$. | import Mathlib
open CategoryTheory Limits Functor Types Function Pullback
theorem Function.isPullback_pulllback {X Y Z : Type u} (f : X → Z) (g : Y → Z) :
IsPullback (C := Type u) (fst (f := f) (g := g)) snd f g := by
sorry | |
0053 | [
"Category"
] | Medium | [
"Limit"
] | [] | problems/0053.md | CAT_statement/S_0053.lean | Theorem: Let $\mathcal{D}$ be a small-complete locally small category, a functor $ G : \mathcal{D}\to \mathcal{C}$ has a left adjoint if and only if $G$ is continuous and for each $c\in \mathcal{C}$, the comma category $(c\downarrow G)$ admits an initial object. | import Mathlib
open CategoryTheory Limits
variable {D : Type u} [Category.{v} D] [HasLimits D] [LocallySmall.{v} D]
variable {C : Type u} [Category.{v} C]
variable (G : D ⥤ C)
theorem has_left_adjoint_iff_continuous_and_initials :
G.IsRightAdjoint ↔ PreservesLimits G ∧ ∀ (c : C), HasInitial (StructuredArrow c G)... | |
0054 | [
"Category"
] | Medium | [
"Limit"
] | [] | problems/0054.md | CAT_statement/S_0054.lean | Theorem: Let $\mathcal{B}$ be a complete category.
Then $\mathcal{B}$ has an initial object if and only if there exists a small set $I$ and an $I$-indexed family of objects $x_i$ such that, for every $s\in \mathcal{B}$, there is an $i\in I$ and an arrow $x_i \to s$. | import Mathlib
open CategoryTheory Limits
variable {B : Type u} [Category.{v} B]
theorem hasInitial_iff_exists_weakly_initial [HasLimits B] :
HasInitial B ↔ ∃ (I : Type v) (x : I → B), ∀ (s : B), ∃ (i : I), Nonempty (x i ⟶ s) := by
sorry | |
0055 | [
"Category"
] | Medium | [
"Limit"
] | [] | problems/0055.md | CAT_statement/S_0055.lean | Theorem: The forgetful functor $U$ : $\mathcal{G}\mathrm{rp}$, $\mathcal{A}\mathrm{b}$, $\mathcal{R}\mathrm{ing}\to \mathcal{S}\mathrm{et}$ creates limits, but they do not preserve coproducts. | import Mathlib
open CategoryTheory Limits
theorem forget_Grp_createsLimits_but_not_coproducts :
Nonempty (CreatesLimits (forget Grp.{u})) ∧
¬ Nonempty (PreservesColimitsOfShape (Discrete Bool) (forget Grp.{u})) := by
sorry
theorem forget_Ab_createsLimits_but_not_coproducts :
Nonempty (CreatesLimits ... | |
0056 | [
"Category"
] | High | [
"Limit"
] | [] | problems/0056.md | CAT_statement/S_0056.lean | Theorem: Let $\mathcal{C}$ and $\mathcal{D}$ be categories and let $G:\mathcal{D}\to \mathcal{C}$ be a functor.
Then $G : \mathcal{D} \to \mathcal{C}$ has a left adjoint if and only if the right Kan extension $\mathrm{Ran}_G \mathrm{Id}_{\mathcal{D}} : C \to D$ exists and is preserved by $G$ (i.e.$G \circ \mathrm{R... | import Mathlib
open CategoryTheory Functor
variable {C : Type u₁} [Category.{v₁} C] {D : Type u₂} [Category.{v₂} D]
theorem hasLeftAdjoint_iff_ran_id_preserved (G : D ⥤ C) :
G.IsRightAdjoint ↔
∃ (R : C ⥤ D) (α : G ⋙ R ⟶ 𝟭 D),
R.IsRightKanExtension α ∧
(R ⋙ G).IsRightKanExtension ((associator ... | |
0057 | [
"Category"
] | Medium | [
"Limit"
] | Exercise 13F, Adámek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0057.md | CAT_statement/S_0057.lean | Theorem: A functor that reflects equalizers (or finite products) reflects isomorphisms. | import Mathlib
open CategoryTheory Limits
variable {C : Type u} [Category.{v} C] {D : Type u'} [Category.{v'} D]
theorem reflectsIsomorphisms_of_reflects_equalizers (F : C ⥤ D)
[ReflectsLimitsOfShape WalkingParallelPair F] : F.ReflectsIsomorphisms := by
sorry
theorem reflectsIsomorphisms_of_reflects_finite_p... |
0058 | [
"Category"
] | High | [
"Limit"
] | [] | problems/0058.md | CAT_statement/S_0058.lean | Definition: Let $\mathcal C$ be a locally small category.
An object $c \in\mathcal C$ is called $\textbf{compact}$ if $\mathrm{hom}_{\mathcal C} (c,-)$ preserves filtered colimits.
Theorem: A topological space $X$ is compact if and only if it is a compact object in the category $\mathcal{O}\mathrm{p}(X)$, the catego... | import Mathlib
open CategoryTheory
namespace CAT_statement_S_0058
universe u
variable (X : Type u) [TopologicalSpace X]
abbrev Op (X : Type u) [TopologicalSpace X] := TopologicalSpace.Opens X
theorem compactSpace_iff_finitelyPresented_top :
CompactSpace X ↔ IsFinitelyPresentable (C := Op X) (⊤ : Op X) := by
... | |
0059 | [
"Category"
] | High | [
"Limit"
] | Exercise 13D(b), Adámek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0059.md | CAT_statement/S_0059.lean | Definition: A functor $F : \mathcal C \to \mathcal D$ is said to $\textbf{lift limits}$ if for every diagram $D: \mathcal I\to \mathcal C$ and every limit $L$ of $F\circ D$, there exists a limit $L'\in\mathcal D$ such that $F(L')\cong L$.
Theorem: There is a functor that lifts limits but is not faithful. | import Mathlib
open CategoryTheory Limits
namespace CAT_statement_S_0059
universe w' w'₁ w w₁ v₁ v₂ v₃ u₁ u₂ u₃
variable {C : Type u₁} [Category.{v₁} C]
variable {D : Type u₂} [Category.{v₂} D]
variable {J : Type w} [Category.{w'} J] {K : J ⥤ C}
structure LiftableCone₂ (K : J ⥤ C) (F : C ⥤ D) (c : Cone (K ⋙ F)) wh... |
0060 | [
"Category"
] | High | [
"Limit"
] | [] | problems/0060.md | CAT_statement/S_0060.lean | Theorem: Suppose $\mathcal{B}$ is locally small, complete, has a small coseparating set $S$, and has the property that every family of subobjects has an intersection. Then $\mathcal{B}$ has an initial object. | import Mathlib
open CategoryTheory Limits
theorem has_initial_of_locally_small_complete_coseparating {ℬ : Type u} [Category.{v} ℬ]
[LocallySmall.{w} ℬ] [HasLimitsOfSize.{w, w} ℬ] {S : Set ℬ} [Small.{w} S]
(hS : IsCoseparating S) (h : ∀ (A : ℬ), ∀ (s : Set (Subobject A)), ∃ (f : Subobject A),
IsGLB s f)... | |
0061 | [
"Category"
] | High | [
"Limit"
] | [] | problems/0061.md | CAT_statement/S_0061.lean | Theorem: Let $\mathcal{C} = \mathrm{Vec}_{\mathbb{k}}$ the category of finite dimensional vector spaces over a field $\mathbb{k}$. Then the coend of the hom functor $\mathrm{Hom} \colon \mathcal{C}^{\mathrm{op}} \times \mathcal{C} \to \mathcal{C}$ is $\mathbb{k}$ equipped with the usual trace maps $\mathrm{Hom}(V,V) \t... | import Mathlib
open CategoryTheory Limits
theorem coend_hom_is_trace_of_matrices
(𝕜 : Type u) [Field 𝕜] :
∀ (F : (ModuleCat 𝕜)ᵒᵖ ⥤ ModuleCat 𝕜 ⥤ ModuleCat 𝕜),
(∀ X Y, (F.obj (Opposite.op X)).obj Y ≅ ModuleCat.of 𝕜 (X →ₗ[𝕜] Y)) →
∃ (T : ModuleCat 𝕜),
(∃ (tr : ∀ X, (F.obj (Opposite.o... | |
0062 | [
"Category"
] | High | [
"Limit"
] | Exercise 12N(b), Adámek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0062.md | CAT_statement/S_0062.lean | Definition: Let $\mathcal C$ be a category. Let $S$ be a family of subobjects $(s_n,i_n)$ of an object $c\in \mathcal C$, indexed by a class $I$.
A subobject $(x,i:x\to c)$ of $c$ is called an $\textbf{intersection}$ of $S$ provided that the following two conditions are satisfied:
(1) $i$ factors through each $i_n$ i... | import Mathlib
open CategoryTheory Limits
namespace CAT_statement_S_0062
universe u v
variable {C : Type u} [Category.{v} C]
def IsIntersectionOf {B : C} (A : Subobject B) (S : Set (Subobject B)) : Prop :=
(∀ Ai, Ai ∈ S → A ≤ Ai) ∧
(∀ A' : Subobject B, (∀ Ai, Ai ∈ S → A' ≤ Ai) → A' ≤ A)
def HasIntersections (... |
0063 | [
"Category"
] | High | [
"Limit"
] | [] | problems/0063.md | CAT_statement/S_0063.lean | Definition: Let $\mathcal C$ be a locally small category.
An object $c \in\mathcal C$ is called $\textbf{compact}$ if $\mathrm{hom}_{\mathcal C} (x,-)$ preserves filtered colimits.
Theorem: For $\mathcal{G}\mathrm{rp}$, an object is compact if and only if it is finitely presented as a group. Every group can be reali... | import Mathlib
open CategoryTheory
namespace CAT_statement_S_0063
universe u
def IsFinitelyPresentedGrp (X : Type u) [Group X] : Prop :=
∃ (α : Type u) (rels : Set (FreeGroup α)), Finite α ∧ rels.Finite ∧ Nonempty (X ≃* PresentedGroup rels)
theorem isCompactObject_Grp_iff_finite_presented (X : Type u) [Group X... | |
0064 | [
"Category"
] | High | [
"Limit"
] | [] | problems/0064.md | CAT_statement/S_0064.lean | Definition: Let $\mathcal C$ be a locally small category.
An object $c \in\mathcal C$ is called $\textbf{compact}$ if $\mathrm{hom}_{\mathcal C} (x,-)$ preserves filtered colimits.
Theorem: Let $A$ be a ring.
For the category of right $A$-modules $\mathcal{A}\mathrm{b}_A$, an object is compact if and only if it... | import Mathlib
open CategoryTheory
universe u v w
theorem isCompactObject_Grp_iff_finite_presented {A : Type u} [Ring A] (X : Type v) [Group X] [AddCommGroup X] [Module A X] : CategoryTheory.IsFinitelyPresentable (ModuleCat.of A X) ↔ Module.FinitePresentation A X := by
sorry
theorem module_realized_as_direct_l... | |
0065 | [
"Category"
] | High | [
"Limit"
] | [] | problems/0065.md | CAT_statement/S_0065.lean | Theorem: Let $\mathcal{C}$ be a complete, wellpowered, cowellpowered and have a separator $s$.
Then $\mathcal{C}$ is cocomplete if and only if for each set $I$, there exists an $I$-th copower of $S$ in $\mathcal{C}$. | import Mathlib
open CategoryTheory Limits
variable {C : Type u} [Category.{v} C]
theorem hasColimits_iff_hasCoprod_of_separator
[HasLimits C]
[WellPowered C]
[WellPowered Cᵒᵖ]
(S : C) (hS : IsSeparator S) :
HasColimits C ↔ ∀ (I : Type v), HasColimit (Discrete.functor (fun (_ : I) => S)) := by
s... | |
0066 | [
"Category"
] | High | [
"Limit"
] | Exercise 12N(c), Adámek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0066.md | CAT_statement/S_0066.lean | Definition: Let $\mathcal C$ be a category. Let $S$ be a family of subobjects $(s_n,i_n)$ of an object $c\in \mathcal C$, indexed by a class $I$.
A subobject $(x,i:x\to c)$ of $c$ is called an $\textbf{intersection}$ of $S$ provided that the following two conditions are satisfied:
(1) $i$ factors through each $i_n$ i... | import Mathlib
open CategoryTheory Limits
namespace CAT_statement_S_0066
universe u v
variable {C : Type u} [Category.{v} C]
def IsIntersectionOf {B : C} (A : Subobject B) (S : Set (Subobject B)) : Prop :=
(∀ Ai, Ai ∈ S → A ≤ Ai) ∧
(∀ A' : Subobject B, (∀ Ai, Ai ∈ S → A' ≤ Ai) → A' ≤ A)
def HasIntersections (... |
0067 | [
"Category"
] | Medium | [
"Limit"
] | [] | problems/0067.md | CAT_statement/S_0067.lean | Theorem: Let $\omega$ be the ordinal of natural numbers.
Consider $F:\omega^{op}\to \mathcal{R}\mathrm{ing}$ with $F_n:=k[x]/(x^n)$ and $f_n:k[x]/(x^{n+1})\to k[x]/(x^n)$.
Then the limit exists and is isomorphic to $k[[x]]$. | import Mathlib
open CategoryTheory Polynomial Limits
universe u
namespace CAT_statement_S_0067
variable (k : Type u) [Field k]
noncomputable def F : Natᵒᵖ ⥤ RingCat :=
{
obj := fun ⟨n⟩ => RingCat.of ((k[X] ⧸ Ideal.span {(X ^ n : k[X])}))
map := fun {A B} f => match A, B with
| ⟨n⟩, ⟨m⟩ => match f wi... | |
0068 | [
"Category"
] | High | [
"Limit"
] | [] | problems/0068.md | CAT_statement/S_0068.lean | Theorem: There is a category $\mathcal{C}$ such that there exists two regular epimorphisms $f:c\to d$ and $g:c'\to d'$ in which the product of $f$ and $g$ is not regularly epic. | import Mathlib
open CategoryTheory Limits
universe u
theorem regular_epimorphism_not_product_regular_epimorphism : ∃ (C : Type (u+1)) (inst : Category C) (c d c' d' : C) (f : c ⟶ d) (g : c' ⟶ d') (inst₁ : RegularEpi f) (inst₂ : RegularEpi g) (hasProd₁ : HasBinaryProduct c c') (hasProd₂ : HasBinaryProduct d d'), IsE... | |
0069 | [
"Category"
] | High | [
"Limit"
] | Exercise 11O, Adámek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0069.md | CAT_statement/S_0069.lean | Theorem: An abelian group is torsion free if and only if it is a directed colimit in $\mathcal{A}\mathrm{b}$ of free abelian groups. | import Mathlib
open CategoryTheory Limits
theorem torsionFree_iff_isFilteredColimit_free
(A : ModuleCat ℤ) :
NoZeroSMulDivisors ℤ A ↔
∃ (J : Type) (_ : SmallCategory J) (_ : IsFiltered J)
(F : J ⥤ ModuleCat ℤ),
(∀ j : J, Module.Free ℤ (F.obj j)) ∧
Nonempty (A ≅ colimit F) := by
... |
0070 | [
"Category"
] | High | [
"Limit"
] | Exercise 13C, Adámek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0070.md | CAT_statement/S_0070.lean | Definition: A concrete category $(\mathcal C, U)$ over $\mathcal B$ is said to $\textbf{have (small) concrete limits}$ if $\mathcal C$ has all small limits and $U$ preserves them.
Theorem: Let $(\mathcal{C},U)$ have small concrete limits.
Then $U$ reflects small limits if and only if $U$ reflects isomorphisms. | import Mathlib
open CategoryTheory Limits
variable {C : Type u} [Category.{v} C]
variable {D : Type u'} [Category.{v'} D]
variable (U : C ⥤ D)
theorem reflects_limits_iff_reflects_isomorphisms_preserves_limits
[HasLimitsOfSize.{v, v} C]
[PreservesLimitsOfSize.{v, v} U]
[CategoryTheory.Functor.Faithful U... |
0071 | [
"Category"
] | Medium | [
"Limit"
] | Exercise 13D(a), Adámek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0071.md | CAT_statement/S_0071.lean | Definition: A functor $F : \mathcal C \to \mathcal D$ is said to $\textbf{lift limits}$ if for every diagram $D: \mathcal I\to \mathcal C$ and every limit $L$ of $F\circ D$, there exists a limit $L'\in\mathcal D$ such that $F(L')\cong L$.
Theorem: A functor that lifts equalizers is faithful if and only if it reflects... | import Mathlib
open CategoryTheory Limits
namespace CAT_statement_S_0071
universe uC vC uD vD w w'
variable {C : Type uC} [Category.{vC} C]
variable {D : Type uD} [Category.{vD} D]
variable (F : C ⥤ D)
variable {J : Type w} [Category.{w'} J]
class LiftsLimit (K : J ⥤ C) (F : C ⥤ D): Prop where
lifts {c : Cone... |
0072 | [
"Category"
] | High | [
"Limit"
] | Exercise 12K, Adámek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0072.md | CAT_statement/S_0072.lean | Theorem: A full subcategory of the category $\mathcal{T}\mathrm{op}^{CH}$ of compact Hausdorff spaces is reflective if and only if it is cocomplete and contains at least one non-empty space. | import Mathlib
open CategoryTheory Limits Topology
universe u
variable {D : Type (u+1)} [Category.{u} D]
variable (i : D ⥤ CompHaus.{u})
variable [CategoryTheory.Functor.Full i] [CategoryTheory.Functor.Faithful i]
theorem reflective_iff_cocomplete_and_contains_nonempty_of_full_subcategory_CompHaus :
Nonempty (C... |
0073 | [
"Category"
] | Medium | [
"Limit"
] | [] | problems/0073.md | CAT_statement/S_0073.lean | Definition: A functor $F : \mathcal C \to \mathcal D$ is said to $\textbf{lift limits}$ if for every diagram $D: \mathcal I\to \mathcal C$ and every limit $L$ of $F\circ D$, there exists a limit $L'\in\mathcal D$ such that $F(L')\cong L$.
Theorem: The forgetful functor $U:\mathcal{T}\mathrm{op}\to \mathcal{S}\mathrm{... | import Mathlib
open CategoryTheory Limits
namespace CAT_statement_S_0073
universe w' w₂' w w₂ v₁ v₂ v₃ u₁ u₂ u₃
variable {C : Type u₁} [Category.{v₁} C]
variable {D : Type u₂} [Category.{v₂} D]
variable {J : Type w} [Category.{w'} J] {K : J ⥤ C}
class LiftsLimit (K : J ⥤ C) (F : C ⥤ D): Prop where
lifts {c : ... | |
0074 | [
"Category"
] | High | [
"Cocompletion"
] | Corollary A.5, Jiří Adámek, Liang-Ting Chen, Stefan Milius, and Henning Urbat, Reiterman's Theorem on Finite Algebras for a Monad, arXiv:2101.00942 | [] | problems/0074.md | CAT_statement/S_0074.lean | Theorem: Let $\mathcal C$ be a small category. A category $\mathcal L$ containing $\mathcal C$ as a full subcategory is an
pro-completion of $\mathcal C$ if and only if the following conditions hold:
(1) $\mathcal L$ has cofiltered colimits,
(2) every object of $\mathcal L$ is the colimit of a cofiltered diagram in $... | import Mathlib
open CategoryTheory Limits
universe u v w u₁ v₁
namespace CAT_statement_S_0074
noncomputable section
abbrev Pro (C : Type u) [Category.{v} C] : Type (max u (v + 1)) := (Ind (Cᵒᵖ))ᵒᵖ
abbrev proYoneda (C : Type u) [SmallCategory C] : C ⥤ Pro C :=
CategoryTheory.opOp C ⋙ (CategoryTheory.Ind.yoneda... |
0075 | [
"Category"
] | High | [
"Cocompletion"
] | Proposition 3.2, Chen Ruiyuan 2021, On sifted colimits in the presence of pullbacks, arXiv:2109.12708 | [] | problems/0075.md | CAT_statement/S_0075.lean | Definition: A category is called sifted if the category of cocones over any finite discrete family of objects in it is connected.
Notation: $\mathrm{Rec}(\mathcal{C})$ := free cocompletion of $\mathcal{C}$ under reflexive coequalizers.
Theorem: For a sifted category with pullbacks $\mathcal C$, $\mathrm{Rec}(\mathc... | import Mathlib
namespace CAT_statement_S_0075
open CategoryTheory Limits
universe u v
namespace CategoryTheory.Limits
open Limits Functor
variable {C : Type u} [Category.{v} C]
variable (C) in
abbrev Psh (C : Type u) [Category.{v} C] : Type (max u (v + 1)) :=
Cᵒᵖ ⥤ Type v
inductive RecObjectPresentation : Ps... |
0076 | [
"Category"
] | High | [
"Cocompletion"
] | Jiří Adámek, Liang-Ting Chen, Stefan Milius, and Henning Urbat, Reiterman's Theorem on Finite Algebras for a Monad, arXiv:2101.00942 | [] | problems/0076.md | CAT_statement/S_0076.lean | Theorem: Let $\mathcal Set^{fin}$ be the category of finite sets and functions. Its pro-completion is the category
$$ \mathrm{Pro}(\mathcal Set^{fin}) = \mathcal Stone$$
of Stone spaces, i.e. compact topological spaces in which distinct elements can be separated by
clopen subsets. Morphisms are the continuous functions... | import Mathlib
open CategoryTheory
universe v u
abbrev Pro (C : Type u) [Category.{v} C] : Type (max u (v + 1)) := (Ind (Cᵒᵖ))ᵒᵖ
theorem pro_fintypecat_equiv_profinite : Nonempty ((Pro (FintypeCat)) ≌ Profinite) := by
sorry |
0077 | [
"Category"
] | High | [
"Cocompletion"
] | Theorem 5.1, Chen Ruiyuan 2021, On sifted colimits in the presence of pullbacks, arXiv:2109.12708 | [] | problems/0077.md | CAT_statement/S_0077.lean | Notation:
$\mathrm{Sind}(\mathcal{C})$ := free cocompletion of $\mathcal{C}$ under small sifted colimits;
$\mathrm{Ind}(\mathcal{C})$ := free cocompletion of $\mathcal{C}$ under small filtered colimits;
$\mathrm{Rec}(\mathcal{C})$ := free cocompletion of $\mathcal{C}$ under reflexive coequalizers.
Theorem:
Let $\ma... | import Mathlib
namespace CAT_statement_S_0077
open CategoryTheory Limits
universe u v
namespace CategoryTheory.Limits
open Limits Functor
variable {C : Type u} [Category.{v} C]
abbrev Psh (C : Type u) [Category.{v} C] : Type (max u (v + 1)) :=
Cᵒᵖ ⥤ Type v
inductive RecObjectPresentation : Psh C → Type (max u... |
0078 | [
"Category"
] | High | [
"Cocompletion"
] | Lemma 6.2, Chen Ruiyuan 2021, On sifted colimits in the presence of pullbacks, arXiv:2109.12708 | [] | problems/0078.md | CAT_statement/S_0078.lean | Definition: For $F : \mathcal{C} \to \mathcal{D}$, we define the induced cocontinuous functor $\mathrm{Lan}_{F^{op}} : \mathcal{P}sh(\mathcal{C}) \to \mathcal{P}sh(\mathcal{D})$, by $\phi\mapsto \phi \star yF$, where $\phi\star yF$ is the $\phi$-weighted colimit of the diagram $yF$ and $y$ is the Yoneda embedding.
No... | import Mathlib
namespace CAT_statement_S_0078
open CategoryTheory Limits Functor
universe u v
namespace CategoryTheory
namespace Limits
variable {C : Type u} [Category.{v} C]
structure SindObjectPresentation (A : Cᵒᵖ ⥤ Type v) where
I : Type v
[ℐ : SmallCategory I]
[hI : IsSifted I]
F : I ⥤ C
ι : F ⋙ ... |
0079 | [
"Category"
] | Easy | [
"Abelian"
] | [] | problems/0079.md | CAT_statement/S_0079.lean | Theorem: Let $\mathcal{A}$ be an additive category.
Let $x,y,z$ be objects in $\mathcal{A}$.
Then the composition $\hom_{\mathcal{A}}(y,z)\times \hom_{\mathcal{A}}(x,y)\to \hom_{\mathcal{A}}(x,z)$ is bilinear map. | import Mathlib
open CategoryTheory
variable {C : Type u} [Category.{v} C] [Preadditive C]
structure IsBilinear {X Y Z : C} (f : (Y ⟶ Z) → ((X ⟶ Y) → (X ⟶ Z))) : Prop where
map_add_left : ∀ (a b : Y ⟶ Z) (g : X ⟶ Y),
f (a + b) g = f a g + f b g
map_add_right : ∀ (a : Y ⟶ Z) (g h : X ⟶ Y), f a (g + h) = f a g... | |
0080 | [
"Category"
] | Easy | [
"Abelian"
] | [] | problems/0080.md | CAT_statement/S_0080.lean | Theorem: Let $\mathcal{A}$ be an abelian category and let $f$ be a morphism in $\mathcal{A}$.
Then $f$ is an isomorphism if and only if $f$ is monic and epic. | import Mathlib
open CategoryTheory
variable {C : Type*} [Category C] [Abelian C]
theorem isIso_iff_mono_and_epi {X Y : C} (f : X ⟶ Y) :
IsIso f ↔ (Mono f ∧ Epi f) := by
sorry | |
0081 | [
"Category"
] | Easy | [
"Abelian"
] | [] | problems/0081.md | CAT_statement/S_0081.lean | Theorem: Let $\mathcal{A}$ be an abelian category and let $f$ be a morphism in $\mathcal{A}$.
Then $f$ is monic if and only if $\ker(f)=0$. | import Mathlib
open CategoryTheory Limits Category
variable {C : Type*} [Category C] [Abelian C]
theorem mono_iff_isZero_kernel {X Y : C} (f : X ⟶ Y) :
Mono f ↔ IsZero (kernel f) := by
sorry | |
0082 | [
"Category"
] | Easy | [
"Abelian"
] | [] | problems/0082.md | CAT_statement/S_0082.lean | Theorem: $\mathbb{k}$ is the unique (up to isomorphism) simple object in $\mathrm{Vect}_{\mathbb{k}}$. | import Mathlib
open Module
variable (𝕜 : Type u) [Field 𝕜]
instance isSimpleModule_self : IsSimpleModule 𝕜 𝕜 := by
constructor
intro N
have : IsSimpleOrder (Submodule 𝕜 𝕜) := by infer_instance
exact eq_bot_or_eq_top N
theorem unique_simple_object (M : Type v) [AddCommGroup M] [Module 𝕜 M] [IsSimpleMo... | |
0083 | [
"Category"
] | Medium | [
"Abelian"
] | [] | problems/0083.md | CAT_statement/S_0083.lean | Theorem: $\mathbb{Z}_p$ is simple object in $\mathcal{A}\mathrm{b}$ when $p$ is prime number. | import Mathlib
open CategoryTheory
variable (p : ℕ) [Fact p.Prime]
theorem ZMod_simple : CategoryTheory.Simple (ModuleCat.of ℤ (ZMod p)) := by
sorry | |
0084 | [
"Category"
] | Medium | [
"Abelian"
] | [] | problems/0084.md | CAT_statement/S_0084.lean | Theorem: $\mathcal{G}\mathrm{rp}$ is not an additive category. | import Mathlib
open CategoryTheory Limits
def IsAdditiveCategory (C : Type u) [Category.{v} C] : Prop :=
∃ (_ : Preadditive C), HasZeroObject C ∧ HasFiniteBiproducts C
theorem Grp_not_is_additive : IsEmpty (IsAdditiveCategory Grp.{u}) := by
sorry | |
0085 | [
"Category"
] | Medium | [
"Abelian"
] | [] | problems/0085.md | CAT_statement/S_0085.lean | Definition: A functor is called $\textbf{left exact}$ if it preserves all finite limits.
Theorem: Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories and $F:\mathcal{A}\to\mathcal{B}$ be a functor. Then $F$ is left exact if and only if $F$ is additive and $F$ maps each exact sequence $0\to x\to y\to z$ to an ex... | import Mathlib
open CategoryTheory Functor Limits ShortComplex
variable {C D : Type*} [Category C] [Category D]
variable [Abelian C] [Abelian D]
theorem preservesFiniteLimits_tfae
(F : C ⥤ D) [F.Additive] : List.TFAE
[
∀ (S : ShortComplex C), S.ShortExact → (S.map F).Exact ∧ Mono (F.map S.f),
∀ (... | |
0086 | [
"Category"
] | Medium | [
"Abelian"
] | [] | problems/0086.md | CAT_statement/S_0086.lean | Theorem: Let $\mathcal{A}$ be an abelian category and let $P\in\mathcal{A}$.
Then $\hom_{\mathcal{A}}(P,-):\mathcal{A}\to \mathcal{A}\mathrm{b}$ is right exact if and only if $\hom_{\mathcal{A}}(P,-):\mathcal{A}\to \mathcal{A}\mathrm{b}$ preserves epimorphism. | import Mathlib
open CategoryTheory Limits Opposite
variable {A : Type u} [Category.{v} A] [Abelian A]
theorem hom_rightExact_iff_preserves_epi (P : A) :
PreservesFiniteColimits (preadditiveCoyoneda.obj (op P)) ↔
Functor.PreservesEpimorphisms (preadditiveCoyoneda.obj (op P)) := by
sorry | |
0087 | [
"Category"
] | High | [
"Abelian"
] | [] | problems/0087.md | CAT_statement/S_0087.lean | Definition: An Abelian category $\mathcal A$ is called $\textbf{semisimple}$ if any short exact sequence in $\mathcal A$ is splittable.
Theorem: Let $\mathcal{A}$ be an abelian category.
Then the followings are equivalent:
\begin{enumerate}
\item $\mathcal{A}$ is semisimple;
\item any obejct i... | import Mathlib
open CategoryTheory Limits
variable {A : Type u} [Category.{v} A] [Abelian A]
def IsSemisimple (A : Type u) [Category.{v} A] [Abelian A] : Prop :=
∀ (S : ShortComplex A), S.ShortExact → Nonempty S.Splitting
theorem isSemisimple_iff_injective_iff_projective :
(IsSemisimple A ↔ ∀ (X : A), Injecti... | |
0088 | [
"Category"
] | Medium | [
"Abelian"
] | [] | problems/0088.md | CAT_statement/S_0088.lean | Theorem: Let $\mathcal{A}$ be an abelian category.
If $x,y$ are simple objects in $\mathcal{A}$. Then each non-zero $f:x\to y$ are isomorphism.
In particular, if $x$ is simple, then $\hom_{\mathcal{A}}(x,x)$ is a division ring; if $x\neq y$ ,then $\hom_{\mathcal{A}}(x,y)=0$. | import Mathlib
open CategoryTheory
variable {𝒜 : Type*} [Category 𝒜] [Abelian 𝒜]
theorem simple_objects_nonzero_morphisms_iso
{x y : 𝒜} [Simple x] [Simple y] (f : x ⟶ y) (h : f ≠ 0) :
IsIso f := by
sorry
theorem simple_object_end_is_division_ring
(x : 𝒜) [Simple x] :
Nonempty (DivisionRing... | |
0089 | [
"Category"
] | Medium | [
"Abelian"
] | [] | problems/0089.md | CAT_statement/S_0089.lean | Definition: Let $\mathcal C$ be a category with zero morphisms. An object $c\in\mathcal C$ is called $\textbf{mono-simple}$ if it has no proper subobjects. An object $c\in\mathcal C$ is called $\textbf{epi-simple}$ if it has no proper quotient objects.
Theorem: Let $\mathcal{A}$ be a pre-additive category in which all... | import Mathlib
open CategoryTheory
class IsSplitMonoCategory (A : Type*) [Category A] where
splitMonoOfMono {X Y : A} (f : X ⟶ Y) [Mono f] : Nonempty (SplitMono f)
class IsSplitEpiCategory (A : Type*) [Category A] where
splitEpiOfEpi {X Y : A} (f : X ⟶ Y) [Epi f] : Nonempty (SplitEpi f)
variable {A : Type*} [Ca... | |
0090 | [
"Category"
] | Medium | [
"Abelian"
] | [] | problems/0090.md | CAT_statement/S_0090.lean | Definition: A category is called $\textbf{normal}$ if each monomorphism is a kernel.
Definition: A category is called $\textbf{conormal}$ if each epimorphism is a cokernel.
Definition: A category is called $\textbf{binormal}$ if it is both normal and conormal.
Definition: Let $\mathcal C$ be a category.
An object $c... | import Mathlib
open CategoryTheory
variable {A : Type*} [Category A] [Limits.HasZeroMorphisms A]
[IsNormalMonoCategory A]
[IsNormalEpiCategory A]
[Limits.HasKernels A]
[Limits.HasCokernels A]
theorem binormal_mono_simple_iff_epi_simple (x : A) :
(∀ (y : A) (f : y ⟶ x) [Mono f], f = 0 ∨ IsIso f) ↔
(∀... | |
0091 | [
"Category"
] | Easy | [
"Monad"
] | [] | problems/0091.md | CAT_statement/S_0091.lean | Theorem: For any monad $(T,\mu,\eta)$ on a category $\mathcal{C}$ and let $\mathcal{C}^T$ be its Elienberg-Moore category.
Let $U:\mathcal{C}^T\to\mathcal{C}$ be the forgetful functor, then it admits a left adjoint. | import Mathlib
open CategoryTheory
variable {C : Type u₁} [Category.{v₁} C]
theorem monad_forget_has_left_adjoint (T : Monad C) :
T.forget.IsRightAdjoint := by
sorry | |
0092 | [
"Category"
] | Medium | [
"Monad"
] | [] | problems/0092.md | CAT_statement/S_0092.lean | Theorem: Let $R$ be a ring. The forgetful functor from the category $\mathcal{M}\mathrm{od}_R$ of $R$-modules to the category $\mathcal{A}\mathrm{b}$ of abelian groups creates all colimits that $\mathcal{A}\mathrm{b}$ admits. | import Mathlib
open CategoryTheory Limits
variable {R : Type u} [CommRing R]
theorem ModuleCat.forgetCreatesColimits :
Nonempty (CreatesColimits (forget₂ (ModuleCat R) AddCommGrp)) := by
sorry | |
0093 | [
"Category"
] | Medium | [
"Monad"
] | [] | problems/0093.md | CAT_statement/S_0093.lean | Theorem: Suppose $\mathcal{C}$ is cocomplete and $G :\mathcal{D} \to \mathcal{C}$ is monadic.
Then $\mathcal{D}$ is cocomplete if and only if $\mathcal{D}$ has coequalizers. | import Mathlib
open CategoryTheory Limits
universe uC uD vC vD w w'
variable {C : Type uC} [Category.{vC} C]
variable {D : Type uD} [Category.{vD} D]
variable (G : D ⥤ C)
theorem cocomplete_iff_hasCoequalizers_of_monadic
[HasColimitsOfSize.{w, w'} C] [MonadicRightAdjoint G] :
HasColimitsOfSize.{w, w'} D ↔ Has... | |
0094 | [
"Category"
] | Easy | [
"Monad"
] | Exercise 20F(b), Adámek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0094.md | CAT_statement/S_0094.lean | Theorem: Let $\mathcal{B}$ be a category and $\mathcal{C}$ be a full reflective subcategory of $\mathcal{B}$. Then the monad associated to the inclusion functor $U : \mathcal{C} \to \mathcal{B}$ is idempotent. | import Mathlib
open CategoryTheory Functor
namespace CAT_statement_S_0094
variable {C : Type*} [Category C]
variable {B : Type*} [Category B]
noncomputable def monadOfRightAdjoint (U : Functor C B) [IsRightAdjoint U] : Monad B :=
(Adjunction.ofIsRightAdjoint U).toMonad
def IsIsoClosed (U : Functor C B) := ∀ (x :... |
0095 | [
"Category"
] | Medium | [
"Monad"
] | [] | problems/0095.md | CAT_statement/S_0095.lean | Theorem: If $\mathcal{D}$ admits coequalizers, a functor $G : \mathcal{D} \to \mathcal{C}$ is monadic if $G$ has a left adjoint, conservative and preserves coequalizers. | import Mathlib
open CategoryTheory Limits
universe u₁ u₂ v₁
variable {C : Type u₁} {D : Type u₂} [Category.{v₁} C] [Category.{v₁} D]
variable {G : D ⥤ C} {F : C ⥤ D} (adjFG : F ⊣ G)
variable [HasCoequalizers D]
variable [G.ReflectsIsomorphisms]
variable [PreservesColimitsOfShape WalkingParallelPair G]
theorem mona... | |
0096 | [
"Category"
] | Medium | [
"Monad"
] | [] | problems/0096.md | CAT_statement/S_0096.lean | Theorem: Consider the adjunction $-\otimes_{\mathbb{Z}}R:\mathcal{A}\mathrm{b}\to \mathcal{A}\mathrm{b}_R$ and $U:\mathcal{A}\mathrm{b}_R\to \mathcal{A}\mathrm{b}$. We obtain a monad $T$.
The $T$-modules are right $R$-modules. | import Mathlib
open CategoryTheory
namespace CAT_statement_S_0096
universe u v
variable (R : Type u) [CommRing R]
abbrev intToR : ℤ →+* R := Int.castRingHom R
noncomputable abbrev U : ModuleCat.{max u v} R ⥤ ModuleCat.{max u v} ℤ :=
ModuleCat.restrictScalars (intToR R)
noncomputable abbrev F : ModuleCat.{max... | |
0097 | [
"Category"
] | Medium | [
"Monad"
] | [] | problems/0097.md | CAT_statement/S_0097.lean | Theorem: Let $\mathcal{C}$, $\mathcal{D}$ be categories and $F : \mathcal{C}\to \mathcal{D}$ be a left adjoint functor to $G: \mathcal{D} \to \mathcal{C}$.
Denote the induced monad of the adjunction $F \dashv G$ by $T := GF$.
Let $K : \mathcal{D} \to \mathcal{C}^T$ be the comparison functor.
If $\mathcal{... | import Mathlib
open CategoryTheory Monad
universe u₁ u₂ v₁
variable {C : Type u₁} [Category.{v₁} C] {D : Type u₂} [Category.{v₁} D]
variable (F : C ⥤ D) (G : D ⥤ C) (adj : F ⊣ G)
theorem comparison_adjunction
[∀ (A : adj.toMonad.Algebra), Limits.HasCoequalizer (F.map A.a) (adj.counit.app (F.obj A.A))] :
∃ K... | |
0098 | [
"Category"
] | Medium | [
"Monad"
] | [] | problems/0098.md | CAT_statement/S_0098.lean | Definition: For any monad $T$ on $\mathcal C$, we define a category $\mathrm{Adj}_T$ whose objects are adjunctions $(F:\mathcal C\to \mathcal D,G,\eta,\epsilon)$ which induce the same monad $T$, and a morphism between $(F:\mathcal C\to \mathcal D,G,\eta,\epsilon)$ and $(F':\mathcal C\to \mathcal D',G',\eta',\epsilon')$... | import Mathlib
open CategoryTheory Monad
namespace CAT_statement_S_0098
variable {C : Type*} [Category C]
structure AdjCat (T : Monad C) where
D : Type*
[category : Category D]
F : Functor C D
U : Functor D C
adj : F ⊣ U
monad_eq : T ≅ Adjunction.toMonad adj
namespace AdjCat
variable {T : Monad C}
in... | |
0099 | [
"Category"
] | Medium | [
"Monad"
] | Exercise 20F(c), Adámek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0099.md | CAT_statement/S_0099.lean | Theorem: The monad associated with the forgetful functor $\mathcal{T}\mathrm{op}\to \mathcal{S}\mathrm{et}$ is idempotent. | import Mathlib
open CategoryTheory
theorem monad_Top_idempotent : IsIso TopCat.adj₁.toMonad.μ := by
sorry |
0100 | [
"Category"
] | High | [
"Monad"
] | [] | problems/0100.md | CAT_statement/S_0100.lean | Theorem: Let $\mathcal{C}$, $\mathcal{D}$ be categories and $F : \mathcal{C} \to \mathcal{D}$ be a left adjoint functor to $G: \mathcal{D}\to \mathcal{C}$.
Denote the induced monad of the adjunction $F \dashv G$ by $T$. The following statements are equivalent:
\begin{enumerate}
\item The comparison fun... | import Mathlib
open CategoryTheory Limits
universe v u u'
namespace CAT_statement_S0100
variable {C : Type u} [Category.{v} C]
variable {D : Type u'} [Category.{v} D]
variable (F : C ⥤ D) (G : D ⥤ C)
variable (adj : F ⊣ G)
abbrev FG : D ⥤ D := G ⋙ F
abbrev K : D ⥤ (adj.toMonad).Algebra :=
Monad.comparison adj
... |
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